Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{-9y^2 + 81y - 180}{y^3 + 4y^2 - 45y}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {-9(y^2 - 9y + 20)} {y(y^2 + 4y - 45)} $ $ r = -\dfrac{9}{y} \cdot \dfrac{y^2 - 9y + 20}{y^2 + 4y - 45} $ Next factor the numerator and denominator. $ r = - \dfrac{9}{y} \cdot \dfrac{(y - 5)(y - 4)}{(y - 5)(y + 9)}$ Assuming $y \neq 5$ , we can cancel the $y - 5$ $ r = - \dfrac{9}{y} \cdot \dfrac{y - 4}{y + 9}$ Therefore: $ r = \dfrac{ -9(y - 4)}{ y(y + 9)}$, $y \neq 5$